1

5 [The value of π is 3.142 correct to three decimal places.]

Sabih Siddiqui 0313-2344852

24

20

125 150

 
2

Diagram I Diagram II

A water tank, shown in Diagram I, is a circular cylinder of radius 24 cm and height 125 cm.
It is open at one end and full of water.

(a) Calculate

(i) the volume, in litres, of water in the tank, [3]

(ii) the total area, in square metres, of the outside of the open tank. [3]

(b) Diagram II shows a rectangular trough of length 150 cm and width 20 cm.
The trough was completely filled with 48 000 cm3 of water from the tank.

H
1
Calculate the depth of the trough. [2]

(c) After the trough had been filled, water started to leak from the tank.
In 2 hours 30 minutes it was found that 20 000 cm3 ran out of the tank.
1

iddiq U
Calculate the rate at which the level of water in the tank was falling.
Express your answer in centimetres per hour. [3]

I
238 126
I
I

92 53

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 2
4

5 [The value of π is 3.142, correct to three decimal places.]

[The volume of a sphere is 4– πr 3.]

Sabih Siddiqui 0313-2344852

3

 
2

Diagram I Diagram II

The diagrams show two ways of packaging 4 identical balls.

The radius of each ball is 3 cm.

Diagram I shows a closed rectangular box with a square base.

Each ball touches the top, the bottom and two sides of the box.
Each ball also touches two other balls.

Diagram II shows a closed cylinder.

The balls touch the ends and the side of the cylinder.

(a) (i) Write down the dimensions of the rectangular box. [1]

H
1
(ii) Calculate the total surface area of the outside of this box. [2]

(b) Calculate the total surface area of the outside of the cylinder. [2]

(c) Calculate the total volume of the 4 balls. 1 [2]

iddiq U
(d) Calculate, correct to three decimal places, the value of

I
volume of the cylinder 238 126
I

––––––––––––––––––– . [2]
volume of the box

(e) Hence state which of the two containers has more space not occupied by the balls. [1]
I

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 3
7

Section B [48 marks]

Sabih Siddiqui 0313-2344852

Answer four questions in this section.

Each question in this section carries 12 marks.

7 [The value of π is 3.142, correct to three decimal places.]

[The surface area of a sphere is 4πr
2
.]
[The volume of a sphere is
43
πr
3.]
9
A closed container is made by joining together
a cylinder of radius 9 cm and a hemisphere

 
of radius 9 cm as shown in Diagram I.
The length of the cylinder is 18 cm. 2
The container rests on a horizontal 18
surface and is exactly half full of water.
Diagram I

(a) Calculate the surface area of the inside of the container that is in contact with the water.
Give your answer correct to the nearest square centimetre. [4]

(b) Show that the volume of the water is 972π cm
3. [2]

(c) The container is held with

its axis vertical, the hemisphere
being at the bottom, as shown
in Diagram II.
Calculate the depth of the water.

H
1

Diagram II [4]

(d) The container is now placed with its

1

iddiq U
circular end on a horizontal
surface as shown in Diagram III.

I
Find the depth of the water. 238 126
I
I

92 53
Diagram III [2]

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 4
6

Section B [48 marks]

Sabih Siddiqui 0313-2344852

Answer four questions in this section.
Each question in this section carries 12 marks.

1
7 [The volume of a pyramid # " base area " height.]
3

 
C
2
N

A B

S R

P Q

The diagram shows a solid traffic bollard.

H
It consists of a square-based pyramid, VABCD, attached to a cuboid,
1 ABCDPQRS.
The vertical line, VNM, passes through the centres, N and M, of the horizontal squares ABCD
and PQRS.

AB # BC # 60 cm and VN # 40 cm.
1

iddiq U
(a) Calculate
(i) VA, [2]

I
238 126
I

(ii) angle VAN, [2]

(iii) angle VAP. [1]

(b) Given also that AP # BQ # CR # DS # 80 cm, calculate

I

92 53
(i) the volume of the bollard, [2]
(ii) the total surface area of the sides and top of the bollard. [3]

(c) The highway authority needs to paint the sides and tops of 17 of these bollards.
The paint is supplied in tins, each of which contains enough paint to cover 8 m2.

Find the number of tins of paint needed. [2]

4024/02 O/N03
 5
7

Section B [48 marks]

Sabih Siddiqui 0313-2344852

Answer four questions in this section.

Each question in this section carries 12 marks.

7 [The surface area of a sphere is 4πr 2.]

1
[The volume of a cone is × base area × height.]
3

[The area of the curved surface of a cone of radius r and slant height l is πrl.]

 
4 2

16

H
1
A drinking glass consists of a hollow cone attached to a solid hemispherical base as shown in the
diagram.
The hemisphere has a radius of 3 cm.
The radius of the top of the cone is 4 cm and the height of the cone is
1 16 cm.

iddiq U
(a) Calculate the total surface area of the solid hemispherical base. [3]

I
(b) Calculate the curved surface area of the outside of the cone. 238 126 [3]
I

(c) (i) The cone contains liquid to a depth of d centimetres.

Giving your reasons, show that the radius of the surface of the liquid is 1 d centimetres. [1]
4
I

(ii) The cone is completely filled with liquid.

92 53
Calculate the volume of the liquid. [2]
(iii) Half of the volume of the liquid from the full cone is now poured out.
Using the answers to parts (i) and (ii), find the depth of the liquid that remains in the cone.
[3]

© UCLES 2004 4024/02/M/J/04 [Turn over

 6
7

Section B [48 marks]

Sabih Siddiqui 0313-2344852

Answer four questions in this section.

Each question in this section carries 12 marks.

1
7 [The volume of a pyramid is 3
× base area × height.]
[The volume of a sphere is 43 πr 3.]

Morph made several different objects from modelling clay.

He used 500 cm3 of clay for each object.

 
(a) He made a square-based cuboid of height 2 cm. 2
Calculate the length of a side of the square. [2]

(b) He made a pyramid with a base area of 150 cm2.

Calculate the height of the pyramid. [2]

(c) He made a sphere.

Calculate the radius of the sphere. [2]

(d) He wrapped the clay around the curved surface 1.5

of a hollow cylinder of height 6 cm.

The thickness of the clay was 1.5 cm.

H
Calculate the radius of the hollow cylinder. 1
6

iddiq U
[4]

I
238 126
I

(e) He made a cone.

Then he cut through the cone, parallel to its

base, to obtain a small cone and a frustum.
I

92 53
The height of the small cone was two-fifths
of the height of the full cone.

Use a property of the volumes of similar

objects to calculate the volume of clay in
the small cone. [2]

© UCLES 2005 4024/02/M/J/05 [Turn over

 7
8

9 [The surface area of a sphere = 4πr 2.]

4
[The volume of a sphere = πr 3.]

Sabih Siddiqui 0313-2344852

3
[The area of the curved surface of a cone of radius r and slant height l is πr l.]
1
[The volume of a cone = × base area × height.]
3

12

 
2

A solid cone has a base radius of 5 cm and height 12 cm.

A solid hemisphere has a radius of 5 cm.
A metal toy is formed by joining the plane faces of the cone and the hemisphere.

(a) Show that the length of the slant edge of the cone is 13 cm. [1]

(b) Calculate

H
(i) the surface area of the toy, [4]
1
(ii) the volume of the toy. [3]

(c) A solid metal cylinder has a radius of 1.5 m and height 2m.
The cylinder was melted down and all of the metal was used to make a large number of these toys.
1
Calculate the number of toys that were made. [4]

iddiq U 238 126

I
I
I

92 53

© UCLES 2005 4024/02/O/N/05

 8
2

Section A [52 marks]

Sabih Siddiqui 0313-2344852

Answer all questions in this section.

1 (a) Solve the equation 3x2 – 4x – 5 = 0, giving your answers correct to two decimal places. [4]

(b) Remove the brackets and simplify (3a – 4b)2. [2]

(c) Factorise completely 12 + 8t – 3y – 2ty. [2]

______________________________________________________________________________________

 
2 (a) A solid cuboid measures 7 cm by 5 cm by 3 cm.
2 3

5
7
(i) Calculate the total surface area of the cuboid. [2]

(ii) A cube has the same volume as the cuboid.

Calculate the length of an edge of this cube. [2]

1–
(b) [The volume of a cone is 3 × base area × height.]
[The area of the curved surface of a cone of radius r and slant height l is π rl.]

H
1
A solid cone has a base radius of 8 cm and a height of 15 cm. 15

Calculate
8
1
(i) its volume, [2]

iddiq U
(ii) its slant height, [1]

I
238 126
I

(iii) its curved surface area, [2]

(iv) its total surface area. [1]

______________________________________________________________________________________
I

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© UCLES 2006 4024/02/M/J/06

 9
6

Section B [48 marks]

Sabih Siddiqui 0313-2344852

Answer four questions in this section.

Each question in this section carries 12 marks.

7 [Surface area of a sphere = 4πr 2]

4
[Volume of a sphere = π r 3]
3
30

 
A hot water tank is made by joining
a hemisphere of radius 30 cm to an 2 70
open cylinder of radius 30 cm and
height 70 cm.

30

(a) Calculate the total surface area, including the base, of the outside of the tank. [4]

(b) The tank is full of water.

(i) Calculate the number of litres of water in the tank. [3]

H
(ii) The water drains from the tank at a rate of 3 litres per second.
1
Calculate the time, in minutes and seconds, to empty the tank. [2]
(iii)

iddiq U
0.6

I
0.3 238 126
I

0.4
I

92 completely fills. 53
All of the water from the tank runs into a bath, which it just
The bath is a prism whose cross-section is a trapezium.
The lengths of the parallel sides of the trapezium are 0.4 m and 0.6 m.
The depth of the bath is 0.3 m.
Calculate the length of the bath. [3]

© UCLES 2006 4024/02/O/N/06

 10
7

Section B [48 marks]

Sabih Siddiqui 0313-2344852

Answer four questions in this section.

Each question in this section carries 12 marks.

7 (a) [The volume of a sphere is 43 πr 3.]

[The surface area of a sphere is 4πr 2.]

 
2
7
20

A wooden cuboid has length 20 cm, width 7 cm and height 4 cm.

Three hemispheres, each of radius 2.5 cm, are hollowed out of the top of the cuboid, to leave the
block as shown in the diagram.

(i) Calculate the volume of wood in the block. [3]

(ii) The four vertical sides are painted blue.
Calculate the total area that is painted blue. [1]
(iii) The inside of each hemispherical hollow is painted white.
The flat part of the top of the block is painted red.
Calculate the total area that is painted

H
(a) white, [2]
1
(b) red. [2]

(b) The volume of water in a container is directly proportional to the cube of its depth.
When the depth is 12 cm, the volume is 576 cm3.
Calculate 1

iddiq U
(i) the volume when the depth is 6 cm, [2]

I
the depth when the volume is 1300 cm3. 238 126
I

(ii) [2]
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 11
6

Section B [48 marks]

Sabih Siddiqui 0313-2344852

Answer four questions in this section.

Each question in this section carries 12 marks.

7 (a) Compost for growing plants consists of 3 parts of soil to 2 parts of sand to 1 part of peat.

(i) Calculate the number of litres of sand in a 75 litre bag of compost. [2]
(ii) Compost is sold in 5 litre, 25 litre and 75 litre bags costing $2, $8.75 and $27 respectively.
Showing your working clearly, state which bag represents the best value for money. [2]

 
(b) [The volume of a cone = 13 × base area × height.] 2
10
The diagram shows a plant pot.
The open end of the plant pot is a circle of radius 10 cm.
12
The closed end is a circle of radius 5 cm.
The height of the plant pot is 12 cm.
The plant pot is part of a right circular cone of height 24 cm. 5

12

(i) Calculate the volume of the plant pot.

Give your answer in litres. [4]
(ii) How many of these plant pots can be completely filled from a 75 litre bag of compost? [2]
(iii) A smaller plant pot is geometrically similar to the original plant pot.
The open end of this plant pot is a circle of radius 5 cm.

H
1
10
5
1

iddiq U 238 a 75 litre bag of126

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I

How many of these plant pots can be completely filled from compost? [2]
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2

Section A [52 marks]

Sabih Siddiqui 0313-2344852

Answer all questions in this section.

1 (a) A flagpole is a cylinder of length 15 m and diameter 14 cm.

Calculate the volume of the flagpole.
Give your answer in cubic metres. [3]

(b) The flagpole, represented by TP in the diagrams below, is hinged at the point P.
It is raised by using two ropes.
Each rope is fastened to the top of the flagpole and the ropes are held at A and B.
The points A, P, B and T are in a vertical plane with A, P and B on horizontal ground.

 
TP = 15 m, AP = 23 m and BP = 12 m.
2
T
(i) When AT̂P = 90°, calculate TP̂A.
15
A B
23 P 12
[2]

T
(ii) When TB̂P = 37°, calculate BP̂T.

15

H
A 1 37° B
23 P 12
[3]

(iii) When the flagpole is vertical, 1 T

iddiq U
calculate the angle of elevation of
the top of the flagpole from A.

I
238 15 126
I

A B
23 P 12
[2]
I

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 13
7

Section B [48 marks]

Sabih Siddiqui 0313-2344852

Answer four questions in this section.

Each question in this section carries 12 marks.

7 A, B, C, D and E are five different shaped blocks of ice stored in a refrigerated room.

(a) At 11 p.m. on Monday the cooling system failed, and the blocks started to melt.
At the end of each 24 hour period, the volume of each block was 12% less than its volume at the start
of that period.
(i) Block A had a volume of 7500 cm3 at 11 p.m. on Monday.

 
Calculate its volume at 11 p.m. on Wednesday. 2 [2]
(ii) Block B had a volume of 6490 cm3 at 11 p.m. on Tuesday.
Calculate its volume at 11 p.m. on the previous day. [2]
(iii) Showing your working clearly, find on which day the volume of Block C was half its volume at
11 p.m. on Monday. [2]

(b) [The volume of a sphere is 43πr 3.]

[The surface area of a sphere is 4πr 2.]
At 11 p.m. on Monday Block D was a hemisphere with radius 18 cm.
Calculate
(i) its volume, [2]
(ii) its total surface area. [2]

H
(c) As Block E melted, its shape was always geometrically similar
1 to its original shape.
It had a volume of 5000 cm3 when its height was 12 cm.
Calculate its height when its volume was 1080 cm3. [2]

iddiq U 238 126

I
I
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 14
9

9 (a) The diagram shows a roll of material.

The material is wound onto a metal

Sabih Siddiqui 0313-2344852

cylinder whose cross-section is a circle
of radius 10 cm.
30
The shaded area shows the cross-section
of the material on the roll.
The outer layer of material forms the 10
curved surface of a cylinder of
radius 30 cm.
200

(i) Calculate, in square centimetres, the area of the cross-section of the material on the roll (shaded
on the diagram). [2]

 
(ii) The material is 200 cm wide on the roll. 2
Calculate, in cubic metres, the volume of the material. [2]
(iii) When unwound, the length of the material is 150 m.
Calculate the thickness of the material, giving your answer in millimetres. [2]

(b) The diagram shows a conical tent.

3
The diameter of the base is 3.5 m and the slant height is 3 m.
It is made from a flat piece of canvas that forms a sector of a circle of 3.5
radius 3 m. The angle at the centre is ␪°.

θ°

H
1

(i) Show that ␪ = 210. [3]

1 of canvas of width w metres.
(ii) As shown, the required shape is cut from a rectangular piece

iddiq U
w

I
238 126
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92 53

Given that w is a whole number, find its least possible value.

Show all your working. [3]

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 15
3

3 (a) A heavy ball hangs from a point P, P

11 m above horizontal ground, by means

Sabih Siddiqui 0313-2344852

of a thin wire.

The point D is on the ground vertically below P. 11

The point B is on the ground 4 m from D.

D B
4
(i) Calculate the angle of elevation of P from B. [2]

 
(ii) The ball swings, with the wire straight,
in the vertical plane PDB.
2 28°
11
X
When the ball is at X, directly above B,
DP̂X = 28°.
D B
Calculate
4
(a) PX, [2]

(b) XB. [3]

4
(b) [The volume of a sphere is 3 πr 3.]

The ball is a sphere of volume 96 cm3.

Calculate its radius. [2]

H
1

iddiq U 238 126

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I
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© UCLES 2009 4024/O2/M/J/09 [Turn over

 16
6

Section B [48 marks]

Sabih Siddiqui 0313-2344852

Answer four questions in this section.

Each question in this section carries 12 marks.

7 (a) A fuel tank is a cylinder of diameter 1.8 m.

(i) The tank holds 25 000 litres when full.
Given that 1m3 = 1000 litres, calculate the length of the cylinder.
Give your answer in metres. [4]
(ii) A

 
2

O
0.9
C E D

B fuel

The diagram shows the cross-section of the cylinder, centre O, containing some fuel.
CD is horizontal and is the level of the fuel in the cylinder.
AB is a vertical diameter and intersects CD at E.
Given that E is the midpoint of OB,
(a) show that EÔD = 60°, [1]

H
(b) calculate the area of the segment BCD, [3]
1
(c) calculate the number of litres of fuel in the cylinder. [2]

4 3
(b) [Volume of a sphere = πr ]
3
A different fuel tank consists of a cylinder of diameter 1.5 m 1

iddiq U
and a hemisphere of diameter 1.5 m at one end. 1.5 1.5

I
238 126
I

The volume of the cylinder is 10 times the volume of the hemisphere.

Calculate the length of the cylinder. [2]
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 17
10
12 (a)
O S p P

Sabih Siddiqui 0313-2344852

T q

R Q

In the diagram, OPQR is a parallelogram.

OP = p and OQ = q.
S is the point on OP such that OS : SP is 1 : 3.
T is the midpoint of OR.

 
2 of p and q,
Giving your answers in their simplest form, find, in terms

(i) QP , [1]

(ii) TS . [2]

(b) In triangle WXY, WX = 24 cm, WY = 17 cm X

and XŴY = 55°. 24

W 55°

17

Y
Diagram I

H
(i) Calculate
1
(a) the area of triangle WXY, [2]

(b) XY. [4]

1 1
(ii) [Volume of a pyramid = 3 × base area × height] V

iddiq U I
The triangle WXY shown in Diagram I forms the 15 X
238 126
I

horizontal base of the triangular pyramid VWXY,

shown in Diagram II. W Z
I

The vertex V is vertically above Z, a point on WX. 92 53

1
WV = 15 cm and WZ = WX.
4 Y

Diagram II
(a) Calculate VZ. [2]

(b) Hence find the volume of the pyramid. [1]

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 18
3

2
A B

Sabih Siddiqui 0313-2344852

120° 70°

D C

The parallelogram ABCD forms part of the pentagon ABCDE.

AB̂C = 70° and BÂE = 120°.

(a) Find

 
2
(i) BĈD, [1]

(ii) EÂD. [1]

(b) ED̂C is twice AÊD.

Find

(i) AÊD, [3]

(ii) ED̂A. [1]

3 The mass and diameter of the planets in the inner solar system are shown in the table.

H
Planet Mass (kg) 1 Diameter (km)
Mercury 3.30 × 1023 4880
Venus 4.87 × 1024 12 100
Earth 5.97 × 1024 1 12 800

iddiq U
Mars 6.42 × 1023 6790

I
238 126
I

(a) List the planets in order of mass, starting with the lowest. [1]

(b) Find the radius, in kilometres, of Mars, giving your answer correct to 1 significant figure. [1]
I

92
(c) Giving your answer in standard form, find the total mass, in kilograms, of Venus and
53 Mars. [1]

(d) [Volume of a sphere = 3 π r 3]

4

Giving your answer in standard form, find the volume, in cubic kilometres, of the Earth. [2]

© UCLES 2010 4024/22/M/J/10 [Turn over

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 19
12

12 [Volume of a cone = 3 π r 2h]

1
[Curved surface area of a cone = π rl]

Sabih Siddiqui 0313-2344852

B
Diagram I shows a solid cone with C
as the centre of its base.
B is the vertex of the cone and A is a point
on the circumference of its base.
AC = 9 cm and BC = 12 cm. Diagram I 12

A C
9

 
2
(a) Calculate

(i) AB, [2]

(ii) the total surface area of the cone, [2]

(iii) the volume of the cone. [2]

(b) The cone in Diagram I is cut, parallel B

to the base, to obtain a small cone shown
in Diagram II and a frustum shown in Diagram II
Diagram III. 3
Y is the centre of the base of the small cone. X Y
X is the point on the circumference of this X 3 Y
base and on the line AB such that Diagram III

H
XY = 3 cm. 1

A C

Calculate 1

iddiq U
(i) BY, [1]

I
238 126
I

(ii) AX, [1]

(iii) the circumference of the base of the small cone, [2]

I

(iv) the volume of the frustum. 92 [2]

53

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been
made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at
the earliest possible opportunity.

University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

© UCLES 2010 4024/22/M/J/10

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 20
6

Section B [48 marks]

Sabih Siddiqui 0313-2344852

Answer four questions in this section.

Each question in this section carries 12 marks.

 
2

20
30

The diagram shows an open rectangular tank with base 20 cm by 30 cm.

The tank contains 9600 cm3 of water.

(a) (i) State the number of litres of water in the tank. [1]

(ii) Calculate the depth of the water. [2]

(iii) Calculate the total surface area of the tank that is in contact with the water. [2]

(iv) The water had entered the tank through a circular pipe of radius 0.8 cm.

H
It flowed through the pipe at 25 centimetres per second. 1

How long did the 9600 cm3 of water take to enter the tank?
Give your answer correct to the nearest second. [3]
1
πr ]
4 3
(b) [Volume of a sphere =

iddiq U
3
250 identical spheres are placed in the bottom of the tank.
Each sphere has a volume of 2.6 cm3.

I
238 126
I

(i) Calculate by how much the water level in the tank will rise.
Give your answer in millimetres. [2]

(ii) Calculate the radius of one of these spheres. [2]

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 21
9
9
B

Sabih Siddiqui 0313-2344852

C A 5
B 5
5
C D
D
Diagram II

 
Diagram I
2
Diagram I shows a cube with a triangular pyramid removed from one vertex.
This triangular pyramid ABCD is shown in Diagram II.
AB = AC = AD = 5 cm.

(a) State the height of this pyramid when the base is triangle ABD. [1]

(b) [The volume of a pyramid = 1 × area of base × height]

3
Calculate

(i) the volume of the pyramid, [2]

(ii) the area of triangle BCD, [3]

(iii) the height of the pyramid when the base is triangle BCD. [3]

(c) An identical triangular pyramid is removed from each of the other 7 vertices of the cube to form

H
the new solid shown in Diagram III. 1

iddiq U 238 126

I
I
I

92 53
Diagram III

The original cube had 6 faces, 8 vertices and 12 edges.

For the new solid, write down the number of

(i) faces, [1]

(ii) vertices, [1]

(iii) edges. [1]

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 22
6

3 (a) Do not
S write in this

Sabih Siddiqui 0313-2344852

margin

 
37°
2
Q

PQRS is a triangular-based pyramid.

RS is perpendicular to the base PQR.
RS = 8 cm and RQ̂ S = 37°.

(i) Find QR.

H
Answer 1.................................. cm [2]

(ii)
N
K 2 1

iddiq U
M
L

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238 126
I

Pyramid KLMN is similar to pyramid PQRS.

MN = 2 cm and the volume of KLMN is 3 cm3.

Find the volume of PQRS.

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Answer ................................. cm3 [2]

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20

10 A cylindrical candle has a height of 5 cm. Do not

12 mm write in this
A is the centre of the top of the candle and B is the

Sabih Siddiqui 0313-2344852

A margin
centre of the base of the candle.
The wick runs from B through A and extends
12 mm above A.
5 cm

(a) How many of these candles can be made using a 2 m length of wick?

 
2

Answer ........................................ [2]

(b) The wick is in the form of a solid cylinder.

The volume of the wick inside the candle from A to B is 0.2 cm3.

(i) Calculate the radius of the wick.

Give your answer in millimetres.

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Answer ................................. mm [3]

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21

(ii) One candle was made by pouring candle wax into a cylindrical mould so that it Do not
surrounded the wick. write in this

Sabih Siddiqui 0313-2344852

This mould has an internal radius of 1.9 cm. margin

(a) Calculate the volume of candle wax required to make this candle.

 
2
Answer ................................. cm3 [3]

(b) How many of these candles can be made using 3 litres of candle wax?

Answer ........................................ [2]

(c)

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length
238 126
I

One of these candles is placed on a rectangular piece of wrapping paper.

The paper is wrapped around the candle so that it covers the outside and there is an
extra 1 cm for an overlap.
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What is the length, in centimetres, of paper required to wrap one candle?

Answer .................................. cm [2]

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20

1 2 Do not
11 [Volume of a cone = π r h]
3 write in this

Sabih Siddiqui 0313-2344852

margin

 
2
The solid above consists of a cone with base radius r centimetres on top of a cylinder
of radius r centimetres.
The height of the cylinder is twice the height of the cone.
The total height of the solid is H centimetres.

(a) Find an expression, in terms of π, r and H, for the volume of the solid.
Give your answer in its simplest form.

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Answer ....................................... [3]
238 126
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(b) It is given that r = 10 and the height of the cone is 15 cm.

(i) Show that the slant height of the cone is 18.0 cm, correct to one decimal place.
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[2]

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15

1
(b) [The volume of a pyramid = 3 × base area × height] Do not
write in this

Sabih Siddiqui 0313-2344852

margin
V

8 Diagram II
B

A F E
D
C

 
The equilateral triangle of side 8 cm in Diagram I forms the base of the triangular pyramid
VABC in Diagram II. 2
The vertex V is vertically above F.
VA = VB = VC = 8 cm.

(i) Calculate the surface area of the pyramid.

Answer ................................cm2 [1]

(ii) Calculate the volume of the pyramid.

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Answer ................................cm3 [3]

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238 1 126
I

(c) A pyramid P is geometrically similar to VABC and its volume is 64 of the volume of VABC.

(i) Find the length of an edge of P.

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Answer ................................. cm [2]

(ii) A pyramid that is identical to P is removed from each of the four vertices of VABC.

State the number of faces of the new solid.

Answer ...................................... [1]

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18

12 [The volume of a sphere = 43 π r3] Do not

write in this

Sabih Siddiqui 0313-2344852

margin

x
 
x
2
A solid consists of a sphere on top of a square-based cuboid.
The diameter of the sphere is x cm.
The base of the cuboid has sides of length x cm.
The sum of the height of the cuboid and one of the sides of the base is 8 cm.

(a) By considering the height of the cuboid, explain why it is not possible for this sphere to
have a radius of 5 cm.

Answer ......................................................................................................................................

............................................................................................................................................. [1]

(b) By taking the value of π as 3, show that the approximate volume, y cm3, of the solid is given by
x3
y = 8x 2 – .
2

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1
[2]

iddiq U
(c) The table below shows some values of x and the corresponding values of y for

I
238 126
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x3
y = 8x 2 – .
2

x 1 2 3 4 5 6 7
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y 7.5 28 96 137.5 180 220.5

(i) Complete the table. [1]

x3
(ii) On the grid opposite, plot the graph of y = 8x 2 – for 1 艋 x 艋 7. [3]
2
(iii) Use your graph to find the height of the cuboid when the volume of the solid is 120 cm3.

Answer ................................. cm [2]

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14

Section B [48 marks] Do not

write in this

Sabih Siddiqui 0313-2344852

Answer four questions in this section. margin

Each question in this section carries 12 marks.

7 (a) Tuna chunks are sold in cylindrical tins.

The 130 g tin costs $1.00 and the 185 g tin costs $1.50.

Which one is the better value for money?

Show all your working.

 
2

Answer ..................................... [2]

(b) A closed cylindrical tin is 11 cm high and the base has a diameter of 7 cm.

11

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1
(i) Calculate the volume of this tin.

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Answer ............................. cm3 [2]

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15

4
[The Surface area of a sphere is 4πr2] [The Volume of a sphere is πr3] Do not
3 write in this

Sabih Siddiqui 0313-2344852

margin
(b) A circular top that can hold 4 hemispherical bowls can be placed on the container.

7 7
21 8

 
Container and Top Top 2 Cross-section

The top is a circle of diameter 21 cm with four circular holes of diameter 7 cm.
A hemispherical bowl of diameter 7 cm fits into each hole.
The cross-section shows two of these bowls.

(i) Calculate the inside curved surface area of one of these hemispherical bowls.

Answer ............................. cm2 [1]

(ii) Calculate the total surface area of the top of the container, including the inside curved

H
surface area of each bowl. 1

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Answer ............................. cm2 [3]

(iii) With the top and the 4 bowls in place, calculate the volume of water required to fill the
I

container. 92 53

Answer ..............................cm3 [3]

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16

9 (a) Shape Iisacylinderwithradius4cmandheighthcm. For

ThevolumeofShape Iis1500cm3. 4 Examiner’s

Sabih Siddiqui 0313-2344852

Use

(i) Findh.

 
Shape I
2

Answer  ............................................... [2]

(ii) Shape Iismadebypouringliquidintoamouldatarateof0.9litresperminute.

Findthenumberofsecondsittakestopourthisliquidintothemould.

H
1
Answer  ................................. seconds[1]

(b) Shape IIisaprismoflength8cmwithatriangularcross-section,

shownshaded.
1
Twosidesoftheshadedtriangleareatrightanglestoeachotherand
havelengths5xcmand12xcm.

iddiq U
12x
GiventhatShape IIalsohasavolumeof1500cm3,findx.

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5x 8

Shape II
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Answer  ............................................... [2]

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(c) Shape IIIisalsoaprismoflength8cmwithatriangularcross-section, For

shownshaded. Examiner’s

Sabih Siddiqui 0313-2344852

Twosidesoftheshadedtriangleareatrightanglestoeachotherand 13y
Use

havelengths5ycmand12ycm.Thethirdsideisoflength13y cm.
12y
y satisfiestheequation4y2+16y–33=0.

(i) Factorise4y2+16y–33.
5y 8

Shape III

 
2
Answer  ............................................... [1]

(ii) Hencesolvetheequation4y2+16y–33=0.

Answer  y=..................or..................[1]

(iii) Findtheareaoftheshadedtriangle.

Answer  ........................................cm2[1]

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1
(iv) FindthetotalsurfaceareaofShape III.

iddiq U 238 126

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Answer  ........................................cm2[3]
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Volume of Shape III
(d) Find asafractioninitssimplestform.
Volume of Shape II

Answer  ............................................... [1]

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22

12 (a) For
Examiner’s
r

Sabih Siddiqui 0313-2344852

Use

46

A cylindrical tank of height 46 cm and radius r cm has a capacity of 70 litres.

Find the radius correct to the nearest centimetre.

 
2

Answer .......................................... cm [3]

(b)

H
1
x
4
125° 20
11 1

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A triangular prism has length 20 cm.

I
The sides of the shaded cross-section are 4 cm, 11 cm and x cm.
238 126
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The angle between the sides of length 4 cm and 11 cm is 125°.

(i) Calculate the area of the shaded cross-section.

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Answer .........................................cm2 [2]

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23

(ii) Calculate the volume of the prism. For

Examiner’s

Sabih Siddiqui 0313-2344852

Use

Answer .........................................cm3 [1]

(iii) Calculate x.

 
2

Answer x = .......................................... [4]

(iv) Calculate the surface area of the prism.

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Answer .........................................cm2 [2]

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18

10 Afueltankerdeliversfuelinacylindricalcontaineroflength9.5mandradius0.8m. For
Examiner’s

Sabih Siddiqui 0313-2344852

(a) Afterseveraldeliveries,thefuelremaininginthecontainerisshowninthediagram. Use

9.5

O
0.8
A
B

 
2
AB ishorizontal,Oisthecentreofthecircularcross-sectionand t = 90c.
AOB

(i) Calculatethecurvedsurfaceareaofthecontainerthatisincontactwiththefuel.

Answer ......................................... m2[2]

(ii) Calculatethevolumeoffuelremaininginthecontainer.

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92 53 m3[4]
Answer .........................................

(iii) Calculatethisvolumeremainingasapercentageofthevolumeofthewholecontainer.

Answer  ...........................................%[2]

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19

(b) Thefuelispumpedthroughacylindricalpipeofradius4.5cmatarateof300cm/s. For

Examiner’s

Sabih Siddiqui 0313-2344852

(i) Calculatethevolumepumpedin1second. Use

 
2
Answer  ....................................... cm3[1]

(ii) Calculatethetimetaken,inminutes,topump25000litresoffuel.
Giveyouranswercorrecttothenearestminute.

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Answer  ................................. minutes[3]

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12

6 (a) A candle is in the shape of a cylinder of radius 1.6 cm and height 7.5 cm.

Sabih Siddiqui 0313-2344852

(i) Calculate the volume of the candle.

Answer ............................................ cm3 [2]

(ii) Six of these candles are packed into a box of height 7.5 cm as shown.

 
2

7.5

(a) Find the length and width of the box.

Answer length = ................................ cm

width = ............................... cm [1]

(b) Calculate the volume of empty space in the box.

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Answer ............................................ cm3 [2]

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14

1 2
9 [Volume of a cone = rr h ]
3

Sabih Siddiqui 0313-2344852

[Curved surface area of a cone = πrl]

15

 
6
2

The diagram shows a solid cone of height 15 cm and base radius 6 cm.

(a) Show that the slant height of the cone is 16.2 cm, correct to one decimal place.

[1]

(b) Calculate the total surface area of the cone.

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Answer ............................................
1 cm2 [3]

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(c) Calculate the volume of the cone.

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Answer ............................................ cm3 [2]

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8

4 (a)
7

Sabih Siddiqui 0313-2344852

3

10

The diagram shows a solid triangular prism. The dimensions are in metres.

(i) Calculate the volume of the prism.

 
2

Answer .......................................m3 [2]

(ii) Calculate the total surface area of the prism.

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Answer .......................................m2 [4]

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7

(b)
S

Sabih Siddiqui 0313-2344852

P L
x° Q
y°
R

 
TwocirclesintersectatLandM. 2
R and P are on the circumference of one circle. S and Q are on the circumference of the other
circle.
PLQ andRLSarestraightlines.
t =x°andMLQ
PLR t = y°.

(i) Completetheproofthat SMQ

t =x°.

Statement Reason

x°= PLR
t =SLQ
t  .............................................................................................
t =SMQ
SLQ t =x° .............................................................................................  [2]

(ii) Provethat PRM

t =y°.

Statement Reason

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 [2]

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(iii) Completethefollowingstatement,givingyourreasons.
238 126
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ThetrianglesPRM andQSM are.............................................

Reasons......................................................................................................................................
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....................................................................................................................................................

....................................................................................................................................................

............................................................................................................................................... [3]

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8

1 2 4
4 [The volume of a cone = πr h] [The volume of a sphere = πr3]
3 3

Sabih Siddiqui 0313-2344852

7.6

4.5

 
2

Asolidisformedbyjoiningaconeofradius4.5cmandheight7.6cmtoahemisphereofradius4.5cm
asshown.

(a) Calculatetheareaofthecirclewheretheyarejoined.

H
1
Answer .....................................cm2[2]

(b) Calculatethetotalvolumeofthesolid.
1

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Answer .....................................cm3[2]

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11 (a)

Sabih Siddiqui 0313-2344852

4
50° 20

Thediagramshowsasolidtriangularprism.
Alllengthsaregivenincentimetres.

(i) Calculatetheareaofthecross-sectionoftheprism.

 
2

Answer .....................................cm2[2]

(ii) Calculatethevolumeoftheprism.

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Answer1 .....................................cm3[1]

(iii) Calculatethetotalsurfaceareaoftheprism.

iddiq U 238 126

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Answer .....................................cm2[5]

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19

(b) Acylinderhasaheightof70cmandavolumeof0.1m3.

Sabih Siddiqui 0313-2344852

Calculatetheradiusofthecylinder,givingyouranswerincentimetres.

        

 
2

Answer ...................................... cm[4]

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8

4 3
4 [The volume of a sphere is rr ]
3

Sabih Siddiqui 0313-2344852

(a)

A spoon used for measuring in cookery consists of a hemispherical bowl and a handle.
The internal volume of the hemispherical bowl is 20 cm3.
The handle is of length 5 cm.

 
2
(i) Find the internal radius of the hemispherical bowl.

Answer .................................... cm [2]

(ii) The hemispherical bowl of a geometrically similar spoon has an internal volume of 50 cm3.

Find the length of its handle.

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Answer .................................... cm [2]

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9

(b) [The surface area of a sphere is 4πr2]

Sabih Siddiqui 0313-2344852

An open hemisphere of radius 5.5 cm is used to make a metal kitchen strainer.
50 holes are cut out of the curved surface.
Assume that the piece of metal removed to make each hole is a circle of radius 1.5 mm.

Calculate the external surface area that remains.

 
2

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Answer .................................. cm2 [3]

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16

8 (a) OAB is a sector of a circle, centre O, radius 6 cm.

A
6

Sabih Siddiqui 0313-2344852

t = 25°.
AOB
O 25°
(i) Calculate the length of the arc AB. 6
B

Answer .................................... cm [2]

(ii) Calculate the area of the sector OAB.

 
2

Answer .................................. cm2 [2]

(b) The sector OAB from part (a) is the cross-section of a slice of cheese.
O 6 A
The slice has a height of 5 cm. 25°
(i) Calculate the volume of this slice of cheese. 6
B
5

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1

Answer .................................. cm3 [1]

1
(ii) Calculate the total surface area of this slice of cheese.

iddiq U 238 126

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Answer .................................. cm2 [3]

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17

(iii) Another 25° slice of cheese has 3 times the height and twice the radius.

Sabih Siddiqui 0313-2344852

Calculate its volume.

 
2

Answer .................................. cm3 [2]

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1
(c) A dairy produces cylindrical cheeses, each with a volume of 800 cm3. r
The height h cm and the radius r cm can vary. h

(i) Express h in terms of r. 1

iddiq U 238 126

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Answer .......................................... [1]

(ii) What happens to the height if the radius is doubled?

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92 53

Answer .................................................................................................................................. [1]

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18

1 2
11 [ Volume of a cone = πr h ]
3

Sabih Siddiqui 0313-2344852

(a)

3.5 r

20

 
2

Solid I

Solid I is a cylinder with a small cylinder removed from its centre, as shown in the diagram.
The height of each cylinder is 20 cm and the radius of the small cylinder is r cm.
The radius of the large cylinder is 3.5 cm greater than the radius of the small cylinder.
The volume of Solid I is 3000 cm3.

(i) Calculate r.

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Answer r = .................................... [4]

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19

(ii) Solid II is a cone with volume of 3000 cm3.

The perpendicular height of the cone is twice its radius.

Sabih Siddiqui 0313-2344852

Which solid is the taller and by how much?

Solid II

 
2

Answer Solid ............ is the taller by .............................. cm [4]

H
(b) The diagram shows a triangular prism of length 24 cm. 1
24
Its cross-section is an equilateral triangle with sides 8 cm.

Calculate the total surface area of the prism.

1

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238 126
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Answer ................................... cm2 [4]

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8

4 3
4 [The volume of a sphere is rr ]
3

Sabih Siddiqui 0313-2344852

[The surface area of a sphere is 4rr 2 ]

0.8

 
2
1.5

3.8

A hemispherical bowl is made of material that is 0.8 cm thick.

The outside rim of the bowl has radius 9 cm.
The bowl is attached to a base which is a solid cylinder, of radius 3.8 cm and height 1.5 cm.

(a) Calculate the surface area of the inside of the hemispherical bowl.

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238 ...................................
Answer 126 cm2 [2]
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9

(b) Calculate the total volume of material used to make the bowl and the base.

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2

Answer ................................... cm3 [5]

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13

1
(c) [The volume of a pyramid is # area of base # perpendicular height]
3

Sabih Siddiqui 0313-2344852

Calculate the total volume of the building.

 
2

Answer ..................................... m3 [2]

(d) Calculate the angle of elevation of T from H.

H
1

Answer .......................................... [3]

1

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12

1 2
8 [Volume of a cone = rr h]
3

Sabih Siddiqui 0313-2344852

[Curved surface area of a cone = rrl ]
4 3
[Volume of a sphere = rr ]
3
[Surface area of a sphere = 4rr2]

 
2
18

The diagram shows solid A which is made from a hemisphere joined to a cone of equal radius.
The hemisphere and the cone each have radius 6 cm.
The total height of the solid is 18 cm.

(a) Show that the slant height, x cm, of the cone is 13.4 cm, correct to 1 decimal place.

H
1

1 [2]

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(b) Calculate the total surface area of solid A.

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Answer ................................... cm2 [3]

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13

(c) Calculate the volume of solid A.

Sabih Siddiqui 0313-2344852

Answer ................................... cm3 [3]

 
(d) Solid A is one of a set of three geometrically similar solids, A, B and C.
The ratio of the heights of solid A : solid B : solid C is22 : 6 : 1.

(i) Calculate the surface area of solid B correct to 3 significant figures.

Answer ................................... cm2 [2]

(ii) Calculate the volume of solid C correct to 3 significant figures.

H
1
Answer ................................... cm3 [2]

iddiq U 238 126

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18

9 (a) The ventilation shaft for a tunnel is in the shape of a cylinder.

The cylinder has radius 0.4 m and length 15 m.

Sabih Siddiqui 0313-2344852

Calculate the volume of the cylinder.

Answer ..................................... m3 [2]

(b) The diagram shows the cross-section of the tunnel.

 
2

O
4.5 110°

A B

The cross-section of the tunnel is a major segment of a circle, centre O.

t = 110°.
The radius of the circle is 4.5 m and AOB

Calculate the area of the cross-section of the tunnel.

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238 .....................................
126 m2 [4]
I

Answer
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9

1 2
(b) [Volume of a cone = rr h]
3

Sabih Siddiqui 0313-2344852

[Curved surface area of a cone = rrl]

9.5

 
2
A cone has height 9.5 cm and volume 115 cm3.

(i) Show that the radius of the base of the cone is 3.4 cm, correct to 1 decimal place.

H
1 [2]

(ii) Calculate the curved surface area of the cone.

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Answer ....................................cm2 [3]

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16

4
9 (a) [Volume of a sphere = rr 3 ]
3

Sabih Siddiqui 0313-2344852

[Surface area of a sphere = 4rr 2 ]

24

 
3
2
The diagram shows lamp A.
It is made in the shape of a cylinder with a hemisphere on top.
The radius of the hemisphere and the radius of the cylinder are both 3 cm.
The total height of the lamp is 24 cm.

(i) Show that the volume of lamp A is 650 cm3, correct to 3 significant figures.

H
1 [4]

(ii) Calculate the curved surface area of lamp A.

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Answer ................................... cm2 [3]

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17

(iii) Lamp B is mathematically similar to lamp A.

The volume of lamp B is 450 cm3.

Sabih Siddiqui 0313-2344852

Calculate the total height of lamp B.

Answer ..................................... cm [2]

(b) The mass of lamp C is 340 g, correct to the nearest 10 g.

 
8 of these lamps are placed in a packing case. 2
The total mass of the packing case and the 8 lamps is 4.2 kg, correct to the nearest 0.1 kg.

Calculate the upper bound of the mass of the packing case when empty.
Give your answer in kilograms.

Answer ..................................... kg [3]

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1
9 [Volume of a pyramid = × base area × height]
3

Sabih Siddiqui 0313-2344852

F

9.5

D C

4.3

 
E
A 6.2 B
2

The diagram shows a pyramid with a rectangular, horizontal base.

Vertex F of the pyramid is vertically above the centre of the base, E.
AB = 6.2 cm and BC = 4.3 cm.
The length of each sloping edge of the pyramid is 9.5 cm.

(a) Show that the height, EF, of the pyramid is 8.72 cm, correct to 3 significant figures.

H
1

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[4]

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(b) Calculate the volume of the pyramid.

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Answer ����������������������������������� cm3 [2]

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16

10

Sabih Siddiqui 0313-2344852

(8 – x)

30°
20
x

The diagram shows a triangular prism.

All lengths are in centimetres.

 
(a) Show that the volume, V cm3, of the prism is given by V = (40x - 5x 2) .
2

[3]

H
(b) On the grid on the next page, draw the graph of V = 40x - 5x 2 for
1 1 G x G 7.
Three of the points have been plotted for you.

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Sabih Siddiqui 0313-2344852

B
14

27°
A 15 C

The diagram shows a triangular prism.

 
AC = 15 cm, BC = 14 cm and angle ACB = 27°. 2
(a) Calculate AB.

AB = ............................................. cm [3]

H
1 cm3.
(b) The length of the prism is p cm and the volume of the prism is 1000

Calculate p.

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p = ................................................... [3]

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8

1
4 [Volume of cone = rr 2 h ]
3

Sabih Siddiqui 0313-2344852

[Curved surface area of a cone = rrl ]

15

95

 
2
8

The diagram shows a gate post.

It is made in the shape of a cylinder with a cone on top.
The cylinder and the cone each have diameter 8 cm.
The height of the cylinder is 95 cm and the height of the cone is 15 cm.

(a) Calculate the volume of the gate post.

H
1
............................................. cm3 [3]

(b) Show that the total curved surface area of the gate post is 2580 cm2, correct to 3 significant figures.
1

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[5]

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14

1
8 [Volume of cone = rr 2 h ]
3

Sabih Siddiqui 0313-2344852

[Curved surface area of a cone = rrl ]

16

15

12

c 45

 
2

The diagram shows a bowl with a circular base.

The curved surface of the bowl is formed by removing a cone with radius 12 cm and height 45 cm from a
larger cone as shown in the diagram.
The radius of the top of the bowl is 16 cm and its height is 15 cm.

(a) Calculate the volume of the bowl.

H
1

............................................. cm3 [3]

1
(b) The slant height of the cone that has been removed is c cm.

iddiq U
Show that c = 46.6, correct to 3 significant figures.

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[2]

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14

8 A birthday cake is in the shape of a cylinder.

There are two layers of cake and one layer of icing.

Sabih Siddiqui 0313-2344852

10 cm
3 cm
12 mm
3 cm

Each layer of cake has radius 10 cm and height 3 cm.

The icing, between the two layers of cake, has radius 10 cm and height 12 mm.

 
(a) Calculate the volume of icing in the birthday cake. 2
Give your answer in cm3.

........................................... cm3 [2]

(b) The top and curved surface of the birthday cake are now covered with chocolate.

Calculate the area of the birthday cake that is covered with chocolate.

H
1

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.......................................... cm2 [3]

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15

(c) Anil has a slice of this chocolate-covered birthday cake.

Sabih Siddiqui 0313-2344852

10.3
x°

7.5

His slice is a prism of height 7.5 cm.

The top of the cake is a sector, radius 10.3 cm and angle x°.
The volume of his slice is 200 cm3.

 
2
Calculate the value of x.

H
x = 1................................................ [3]

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18

1
10 [Volume of pyramid = 3 # base area # height]

Sabih Siddiqui 0313-2344852

E

12

B C

F 6

  D2
A 10

ABCDE is a rectangular-based pyramid.

AC and BD intersect at F.
EF is perpendicular to FC.

AD = 10 cm, DC = 6 cm and EC = 12 cm.

(a) Show that EF = 10.5 cm, correct to 1 decimal place.

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1

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(b) Find the volume of the pyramid. 238 126
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.......................................... cm3 [2]

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10

5
H

Sabih Siddiqui 0313-2344852

D
G

2.25 E
C

  1.85 2 F
A
2.10
1.55
B

The diagram shows a garden shed positioned on horizontal ground.

It is in the shape of a prism with trapezium ABCD as its cross-section.
The base of the shed, ABFE, is a rectangle.
AB = 1.55 m, AD = 2.25 m, BC = 1.85 m and BF = 2.10 m.

(a) Calculate the volume of the shed.

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1

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............................................ m 3 [3]
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(b)
Solid A

Sabih Siddiqui 0313-2344852

6
40°

 
2
The cross-section of solid A is the sector of a circle of radius 6 cm and angle 40°.
The height of solid A is 5 cm.

(i) Calculate the total surface area of solid A.

H
1
.......................................... cm 2 [4]

(ii) Solid B is mathematically similar to solid A.

The ratio volume of solid A : volume of solid B = 27 : 1.1

iddiq U
Calculate the surface area of solid B.

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.......................................... cm 2 [2]
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4
4 (a) [Volume of a sphere = rr 3 ]
3

Sabih Siddiqui 0313-2344852

[Surface area of a sphere = 4rr 2 ]

16

 
The diagram shows a solid formed by joining a cylinder 2 to a hemisphere.
The diameter of the cylinder is 9 cm and its height is 16 cm.

(i) The volume of the hemisphere is equal to the volume of the cylinder.

Show that the radius of the hemisphere is 7.86 cm, correct to 2 decimal places.

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1

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[4]

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(ii) Calculate the total surface area of the solid.
238 126
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.......................................... cm 2 [3]

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9

(b) A different solid is in the shape of a cuboid.

The cuboid measures 8 cm by 4 cm by 6 cm.

Sabih Siddiqui 0313-2344852

These measurements are given correct to the nearest centimetre.

Calculate the lower bound of the volume of the cuboid.

.......................................... cm 3 [2]

 
2

H
1

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1 2
9 [Volume of a cone = rr h]
3

Sabih Siddiqui 0313-2344852

[Curved surface area of a cone = rrl ]

 
A cone has radius 6 cm and slant height l cm. 2
The total surface area of the cone is 84r cm2.

(a) Show that l = 8.

[2]

(b) Calculate the volume of the cone.

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1

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.......................................... cm3 [3]
238 126
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(c) A similar cone has a total surface area of 47.25r cm2.

Find the radius of this cone.

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............................................ cm [2]

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18

10 (a) A cuboid measures 6.2 cm by 4.8 cm by 2.5 cm.

Each measurement is given correct to the nearest millimetre.

Sabih Siddiqui 0313-2344852

Calculate the upper bound of the surface area of the cuboid.

.......................................... cm2 [3]

 
1
(b) [Volume of a pyramid = # base area # height] 2
3

19

17
D C

X
A B

The diagram shows a square-based pyramid ABCDE.

H
Vertex E is vertically above X, the centre of the square base. 1
The height of the pyramid, EX, is 17 cm.
EC = 19 cm.

(i) Show that the length of the base is 12 cm. 1

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[4]

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(ii) Calculate the volume of the pyramid.

Sabih Siddiqui 0313-2344852

.......................................... cm3 [2]

(iii) Calculate angle CBE.

 
2

Angle CBE = ................................................ [3]

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1

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1
8 [Volume of cone = rr 2 h ]
3

Sabih Siddiqui 0313-2344852

[Curved surface area of a cone = rrl ]

l h

 
2
The diagram shows a paper cup in the shape of a cone.
The diameter of the top of the cup is 7 cm.
The volume of the cup is 110 cm 3 .

(a) Show that the height of the cup, h cm, is 8.57 correct to 2 decimal places.

H
1

1 [3]

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(b) Calculate the slant height, l cm, of the cup.

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l = ................................................. [2]

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(c)
A 7

Sabih Siddiqui 0313-2344852

NOT TO
SCALE

x°
O O

The cup is cut along the line OA.

 
It is opened out into a sector of a circle with centre O 2
and sector angle x°.

Calculate the value of x.

H
1

iddiq U
x = ................................................. [4]

(d) A second paper cup is mathematically similar to the cup with volume 110 cm 3 .

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The volume of the second cup is 165 cm 3 . 238 126
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Calculate the diameter of the top of the second cup.

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............................................ cm [2]

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8

4 (a) A cuboid has dimensions x cm by x cm by 10 cm.

The volume of the cuboid is 62.5 cm 3 .

Sabih Siddiqui 0313-2344852

Find the value of x.

x = ................................................. [2]

(b)

 
O 2
84°

15 NOT TO
SCALE

A
B

A piece of card, AOB, is a sector of a circle, centre O, with angle 84° and radius 15 cm.

(i) Show that the arc length of the sector is 7r cm.

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[1]
238 126
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(ii) OA is joined to OB to form the curved surface of a cone.

Calculate the radius of the cone.

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............................................ cm [2]
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6

4 (a)
12

Sabih Siddiqui 0313-2344852

NOT TO
SCALE
15
9

The diagram shows a pentagon.

All the lengths are in centimetres.

 
(i) Calculate the area of the pentagon. 2

.......................................... cm 2 [2]

(ii) Find the perimeter of the pentagon.

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1

1............................................ cm [3]

iddiq U
4
(b) [Volume of a sphere = rr 3 ]
3

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A sphere has a volume of 2572 cm 3 . 238 126
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Find the radius of the sphere.

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92 53

............................................ cm [3]
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1
(b) [Volume of a pyramid = # base area # height]
3

Sabih Siddiqui 0313-2344852

F

14

E
O
C

 
A
2
B

The regular pentagon ABCDE forms the base of a pyramid.

The vertex F is vertically above O.
The length of each sloping edge of the pyramid is 14 cm.

Calculate the volume of the pyramid.

H
1

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cm 3 [5]
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.........................................
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4
9 [Volume of a sphere = rr 3 ]
3

Sabih Siddiqui 0313-2344852

[Surface area of a sphere = 4rr 2 ]

 
2
20

The diagram shows a wooden bowl.

It is made in the shape of a large hemisphere with a small hemisphere removed from the centre.
The diameter of the large hemisphere is 20 cm.
The width of the rim of the bowl is 2 cm.

(a) Show that the total surface area of the bowl is 364r cm 2 .

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1

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3

1
(d) [Volume of pyramid = # base area # height]
3

Sabih Siddiqui 0313-2344852

x

 
The diagram shows a pyramid with a square base of side length 9 cm.
The pyramid has height x cm and volume y cm 3 . 2

(i) Show that the equation for the volume of the pyramid is y = 27x .

[1]

(ii) By drawing a suitable straight line on the grid on page 2, find the height of the pyramid when
the pyramid and the cuboid have the same volume.

H
1

iddiq U
............................................ cm [3]

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(b)

Sabih Siddiqui 0313-2344852

NOT TO
SCALE
8
75°

A piece of card is a sector of a circle with sector angle 75° and radius 8 cm.

(i) Find an expression, in terms of r, for the arc length of the sector.
Give your answer in its simplest form.

 
2

............................................ cm [2]
1
(ii) [Volume of a cone = rr 2 h]
3

H
The piece of card forms the curved surface area of a cone.1
The cone is filled to the top with water.

Calculate the volume of water in the cone.

1

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......................................... cm 3 [5]

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